Effect of Zeeman coupling on the Majorana vortex modes in iron-based topological superconductors

In the superconducting regime of ${\mathrm{FeTe}}_{(1-x)}{\mathrm{Se}}_x$, there exist two types of vortices which are distinguished by the presence or absence of zero-energy states in their core. To understand their origin, we examine the interplay of Zeeman coupling and superconducting pairings in three-dimensional metals with band inversion. Weak Zeeman fields are found to suppress intraorbital spin-singlet pairing, known to localize the states at the ends of the vortices on the surface. On the other hand, an orbital-triplet pairing is shown to be stable against Zeeman interactions, but leads to delocalized zero-energy Majorana modes which extend through the vortex. In contrast, the finite-energy vortex modes remain localized at the vortex ends even when the pairing is of orbital-triplet form. Phenomenologically, this manifests as an observed disappearance of zero-bias peaks within the cores of topological vortices upon an increase of the applied magnetic field. The presence of magnetic impurities in ${\mathrm{FeTe}}_{(1-x)}{\mathrm{Se}}_x$, which are attracted to the vortices, would lead to such Zeeman-induced delocalization of Majorana modes in a fraction of vortices that capture a large enough number of magnetic impurities. Our results provide an explanation for the dichotomy between topological and nontopological vortices recently observed in ${\mathrm{FeTe}}_{(1-x)}{\mathrm{Se}}_x$.

Figure 1

Phase boundaries between regions with superconducting order parameters ${\mathrm{\Delta }}_1$ or ${\mathrm{\Delta }}_0$ as a function of the ratio $U/V$ of interaction strengths of each channel and the field magnitude ${\mathrm{\Delta }}_Z$ (red solid curve). The blue dashed line is a guide for the eye of the case when ${\mathrm{\Delta }}_Z=0$. The parameters of the Hamiltonian are $m=-0.5$ and $\epsilon =0.5$. Increased Zeeman coupling results in larger regions where interorbital-triplet pairing is the ground state.

Figure 2

Dependence of the vortex-mode energies on the chemical potential $\mu$ obtained from the 3D lattice model with periodic boundary conditions along the $z$ direction for $k_z=0$. (a) Intraorbital-singlet pairing ${\mathrm{\Delta }}_1=0.4$ for ${\mathrm{\Delta }}_Z=0$. (b) Interorbital-triplet pairing ${\mathrm{\Delta }}_0=0.4$ and ${\mathrm{\Delta }}_Z=0.2$. The parameters for the model are $M=4.5,m_0=-2.0$, and $t=1.0$ with the calculation performed on a $48\times 48$ lattice in the $x\text{−}y$ plane. We observe a zero-energy mode state exists for all chemical potentials inside the conduction band when the pairing is interorbital-triplet type.

Figure 3

Spatial profile of the lowest-energy vortex mode in 3D slab geometry for different values of parameter $\alpha$, which controls the amplitude of the interorbital-triplet pairing ${\mathrm{\Delta }}_0=\alpha {\mathrm{\Delta }}_0^0$, intraorbital $s$-wave ${\mathrm{\Delta }}_1=(1-\alpha ){\mathrm{\Delta }}_1^0$, and Zeeman field ${\mathrm{\Delta }}_Z=\alpha {\mathrm{\Delta }}_Z^0$ around the vortex. The calculation is performed on a $24\times 24\times 24$ lattice. The parameters used are $\mu =1.1,{\mathrm{\Delta }}_0^0=0.4,{\mathrm{\Delta }}_1^0=0.4,{\mathrm{\Delta }}_Z^0=0.2,M=4.5,m_0=-2.0$, and $t=1.0$.

Figure 4

$k_z$ momentum dependence of the vortex modes for (a) intraorbital $s$-wave pairing ${\mathrm{\Delta }}_1=0.4$ and (b) interorbital-triplet pairing ${\mathrm{\Delta }}_0=0.4$ with ${\mathrm{\Delta }}_Z=0.2$ and the chemical potential set to be $\mu =1.1$. Additional parameters of the model are the same as in Fig. 2.